One angle decides contact

Raup’s model builds a coiled shell from one generating circle and three numbers: W, how much the whorl expands each turn; D, how far the circle sits from the coiling axis; and T, how far it walks along the axis each turn. The line has been located in that space where successive whorls stop touching to the last bit a double holds, measured what a spire does to it, and found that two ways of measuring the same boundary disagree about whether it is sharp.

Every one of those results was drawn in the parameter plane. None of them says what a person holding a shell would have to look at to find out which side of the line it is on, and the answer turns out to be written in a single angle.

The section that holds the axis #

Cut a shell with the plane that contains its coiling axis and each whorl appears as a disc, once on each side. In the model’s own units the earliest disc has its centre (1 + D)/2 from the axis and T above the apex, and its radius is (1 − D)/2. Every later disc is that one multiplied by W about the apex, once per whorl.

That last fact is the whole of the geometry. A multiplication about a point preserves every angle seen from that point, so whatever angle the first disc subtends at the apex, every disc subtends.

The angle every whorl subtends #

The disc’s centre lies at a distance of √((1 + D)²/4 + T²) from the apex, and the disc’s radius over that distance is the sine of its half-angle. So each whorl subtends

γ = asin( (1 − D) / √((1 + D)² + 4T²) )

about the line through the disc centres, which runs at an angle β from the axis with tan β = (1 + D)/2T. At the shell in the figure, where W = 2.4, D = 0.3 and T = 1, γ is 17.065° and β is 33.024°.

Read off the rings the drawing code actually produces, rather than from the formula, the half-angle does not move from one whorl to the next by more than 1.1 × 10⁻¹⁶ of a radian — the last bit a double holds — on three shells at three expansions each.

Why the answer had to be an angle #

A shell that is a multiplication about its apex looks the same at every scale seen from that point, so whatever decides whether its whorls touch cannot be a length: a length changes by W from one whorl to the next while the verdict does not. It has to be built from quantities the multiplication leaves alone, and those are angles seen from the apex, and W itself.

The rule is the simplest thing that can be built from those — one angle against one function of W — which is some reason to have expected it rather than only to have found it. It is also a reason to distrust any summary of a shell’s form that is a length on its own.

Contact in one inequality #

Two successive discs on the same side of the axis have centres (W − 1) times the first centre’s distance apart, and radii that add to (W + 1) times the first radius. They overlap when the first quantity is the smaller. Divide both by the first centre’s distance and the condition becomes

sin γ > (W − 1) / (W + 1)

which is the whole of the contact condition. Two of Raup’s numbers, D and T, enter only through the angle. The third, W, enters only through the threshold the angle has to clear.

Turned round, a shell whose tube subtends γ touches itself at every expansion below (1 + sin γ)/(1 − sin γ), and at none above it. For the shell in the figure that expansion is 1.8307, so at 2.4 its whorls run free.

The rule, worked for one shell #

For the shell in the first figure, (1 + D)² is 1.69 and 4T² is 4, so the root in the denominator is √5.69 = 2.38537, and sin γ is 0.7/2.38537 = 0.29345 — an angle of 17.064°. The threshold at W = 2.4 is 1.4/3.4 = 0.41176.

The sine is below the threshold, so the whorls run free, and they would touch at any expansion below (1 + 0.29345)/(1 − 0.29345) = 1.29345/0.70655 = 1.8307. Everything about contact for that shell is in those two lines of arithmetic, and none of it needed the shell to be drawn.

Checked against the drawn discs #

An inequality derived in three lines deserves a check that does not share its algebra. The discs themselves are one: compare the distance between two drawn centres with the sum of two drawn radii, with no angle anywhere in the computation.

Over 400,000 random shells — W spread evenly in its logarithm from 1.02 to 20, D from 0 to 0.999 and T from 0 to 3 — the angle rule and the discs disagree 0 times, and 52,827 of the shells, 13.21 per cent, are in contact. At seventeen boundaries already located by bisection, the angle sits on its threshold to within 1.67 × 10⁻¹⁶.

The 13.21 per cent is not a fact about shells. It is a share of the box the random shells were drawn from, and the share of Raup’s cube any geometry claims moves with where the box’s edges are put. The number from this check that does not depend on the box is the zero.

It contains the lines already drawn #

A rule that contained less than the earlier results would be a different rule, so it has to reduce to them, and it does.

With no translation the angle is asin((1 − D)/(1 + D)), and the flip expansion simplifies to 1/D. That is the hyperbola D = 1/W read the other way: at D = 0.2 the rule gives 5.000000000000, at 0.42 it gives 2.380952380952 and at 0.6 it gives 1.666666666667, each equal to the reciprocal to the last printed digit.

With the opening against the axis, D = 0, the angle at the translation that frees every axis distance, √W/(W − 1), is exactly the contact angle: 24.316° at W = 2.4, where that translation is 1.106567, and 45.585° at W = 6, where it is 0.489898. Squaring the angle condition and clearing denominators gives back the quadratic in D the earlier boundary was derived from, term for term.

The plane of angle and expansion #

The threshold angle rises with the expansion and never reaches a right angle: 5.216° at W = 1.2, 13.342° at 1.6, 24.316° at 2.4, 31.588° at 3.2, 39.521° at 4.5, 48.190° at the golden 6.854, and 64.791° at 20.

A shell is a horizontal line on that plane, because its angle is fixed by D and T and does not move with W. The line crosses the threshold curve once. At D = 0.3 and T = 1 the angle is 17.065° and the crossing is at W = 1.8307; at D = 0.1 and T = 2 it is 12.530° and 1.5541; at D = 0.42 and T = 0.5 it is 19.509° and 2.0028.

An apical angle does not know W #

The line through the disc centres and the two tangent lines at β ± γ are all fixed by D and T. Nothing in them depends on the expansion, so the envelope of a spire — the outline a photograph of its side shows — has an apical angle of 2(β + γ) whatever W is.

At D = 0.3 and T = 1 that angle is 100.178° at W = 1.3, at W = 2.4 and at W = 6, with no difference between the three to the last bit. The first of those shells has whorls in contact, and the other two are free.

An outline does not even give the angle #

It is worse than that, because the envelope carries the sum β + γ and contact wants γ alone. Different openings at different heights can share the sum and split it differently. Holding the apical angle at 100.178°, a shell with its opening at D = 0.1 reaches it at a translation of 1.0467 with a tube angle of 22.369°; at D = 0.2 the translation is 1.0234 and the angle 19.706°; at D = 0.42 they are 0.9720 and 13.942°.

Those four shells — the three and the one in the first figure — have one outline angle between them and four different flip expansions: 2.2288, 2.0174, 1.8307 and 1.6348. Grow all four at W = 2 and the first two touch themselves while the last two run free. A photograph of the side of a spire therefore fails twice over: it gives no W, and it does not give the angle W has to be compared with.

Two drawings with one outline and opposite verdicts #

At D = 0.2 and T = 0.6 a whorl subtends 28.126° about a centre line at exactly 45°, and the envelope’s apical angle is 146.251°. Expanding by 1.5 a whorl, the shell’s successive discs run into one another all the way up the section. Its flip expansion is 2.7836, well above the 1.5 it is drawn at.

Expanding by 3 instead, the same opening at the same place between the same two lines leaves every disc clear of the next. A person comparing the two outlines, or measuring the apical angle on each, would find the same number twice — and one of the shells touches itself and the other does not.

What a silhouette does carry #

That needs one qualification, because a side view of a spire is not only its envelope. The scallops along the edge of a silhouette are the whorls’ sections at the extreme azimuth, and their positions along the cone are geometric with ratio W.

So a silhouette with its sutures resolved does carry the expansion, in the spacing of its scallops. What carries no expansion is the angle of the envelope — the one number a spire is usually summarised by, and the one most easily measured off a photograph.

A plan carries no translation #

Look down the coiling axis instead and the outer wall is a logarithmic spiral whose growth factor is W exactly, whatever the height. Handed to the fit that recovers a growth factor, the plan of a shell at W = 3.2 and D = 0.3 returns 3.200000000000 with no translation, at T = 1 and at T = 3, and D = 0.3 each time.

The last whorls of those three shells sit at heights of 0, 58.6 and 175.9 along the axis, and none of it reaches the points. A plan carries W and D and throws T away, which is also why the centre a fit needs is a problem about the plan’s own coordinates and never about the spire.

An axial section carries all three #

An axial section loses nothing. On one side of the axis the discs’ distances from the apex grow by exactly W from each whorl to the next — 2.400000000000 four times over on the shell in the first figure — and any single disc gives the other two numbers. D is the disc’s inner edge over its outer edge, measured from the axis, or (r − ρ)/(r + ρ); T is its height over its outer edge, z/(r + ρ).

From the section drawn at W = 2.4, D = 0.3 and T = 1, those readings return 0.3000 and 1.0000. The section is the one picture of the three in which the contact question can be answered without borrowing a number from somewhere else.

Three numbers out of two discs, by hand #

The disc one whorl out from the apex on that shell has its centre 2.4 × 0.65 = 1.56 from the axis, a radius of 2.4 × 0.35 = 0.84, and a height of 2.4 × 1 = 2.4. Its inner edge is therefore 0.72 from the axis and its outer edge 2.4. D is 0.72/2.4 = 0.3, and T is the height over the outer edge, 2.4/2.4 = 1.

The next disc on the same side is the same numbers multiplied by 2.4: centre 3.744, radius 2.016, height 5.76. Its distance from the apex is √(3.744² + 5.76²) = 6.870 against the first disc’s √(1.56² + 2.4²) = 2.862, and the ratio is 2.400. Two discs and a ruler return all three of Raup’s numbers, which is the practical content of the section carrying them.

How well the angle has to be measured #

The flip expansion is (1 + sin γ)/(1 − sin γ), and its rate of change with the angle is 2 cos γ/(1 − sin γ)². At the shell in the first figure that is 2 × 0.9560/0.4992, or 3.830 for each radian — 0.067 of an expansion for each degree.

So a section read to within a degree places the flip within about seven hundredths of W, and a shell whose expansion sits more than that from its flip has a verdict the measurement cannot overturn. The shell in the figure, at 2.4 against a flip of 1.8307, subtends 17.065° where the contact angle at 2.4 is 24.316°, so it is more than seven degrees of angle clear of the boundary. The rate grows as the angle does — the same derivative is 0.210 of an expansion a degree at a tube angle of 40° — so a wide tube needs a section read more carefully than a narrow one.

What rising does to the threshold #

Hold D and let the spire rise, and the angle falls, because the disc centres move away from the apex while the discs keep their size. At D = 0.42 the flip expansion is 2.3810 with no translation, 2.2534 at T = 0.25, 2.0028 at 0.5, 1.6194 at 1 and 1.3165 at 2.

The curves are steepest where the shells drawn so far sit. At D = 0.3 the flip falls from 3.3333 with no translation to 1.3135 at T = 2.5, and at D = 0.1 it falls from 10.0000 to 1.4266. A spire is, in these terms, a device for lowering the expansion at which a coiled tube starts running into itself.

The boundary as a line of constant angle #

Hold W instead and ask, for each axis distance, how much translation it takes to bring the tube’s angle down to the contact angle. At W = 2.4 that angle is 24.316°, and the translation is 1.106567 with the opening against the axis, 0.944371 at D = 0.1, 0.763985 at 0.2, 0.547723 at 0.3 and 0.202031 at 0.4. Beyond D = 0.4167 there is none, because the tube’s angle is already below the threshold with no translation at all.

Those five numbers are the boundary the spire essay located, point for point. Every point on it is one angle, so the curve in the parameter plane is a contour of a single quantity. The shell in the first figure, at D = 0.3 and T = 1, is past 0.547723 and so free, which is what its section shows.

Why the translation enters as its square #

The earlier measurement found that a small translation moves the boundary in proportion to its square, and the angle says why in one line. The translation appears in γ only as 4T² under a square root, so for small T the sine of the angle falls from (1 − D)/(1 + D) by an amount proportional to T², and the boundary moves with it.

A quantity that enters as a square has a flat start. A shell that has only begun to walk along its axis has changed its angle by almost nothing, which is the geometric content of the finding that a low spire buys almost no clearance.

A slow shell at the same station #

Lower the expansion of the first shell from 2.4 to 1.3 and change nothing else, and every disc overlaps the one a whorl above it. The angle is still 17.065° and the envelope is still 100.178°; the threshold at 1.3 is only 7.5°, well below what the tube subtends.

Nothing about the silhouette’s angle changed. The only change is in the one number a side outline does not carry, and it is the number that decided.

What four hundred thousand agreements are #

It is worth being exact about what the check establishes. The agreements are between an inequality and a computation on the same model, so they establish that the algebra is right for Raup’s shell — a circle multiplied about a point. They say nothing about whether a real shell is that object, and they could not.

What this does not establish #

The rule is for a circular generating curve. A real opening is an oval or a slot, and in ammonites a lobed outline, and whether the angle still decides contact for those is a separate measurement — taken up for the fourth number, the shape of the opening, where the answer is partly yes and partly no.

It says nothing about which angles real shells have, or why. And it assumes the shell is an exact multiplication about its apex, which a shell whose expansion changes as it grows is not.

What would withdraw it #

One shell, among any the drawing code produces, whose rings touch where the sine of their angle is below (W − 1)/(W + 1), or clear where it is above. Or a ring whose half-angle from the apex differs from its neighbours’ by more than arithmetic, which would mean the shell is not a multiplication about one point and the argument above has no object. Both are checked every time the measurement runs.

Whether the ellipse is the circle stretched #

The angle argument used one property of the circle: that its extent along the radius and its extent along the axis are set by one number. An elliptical opening S times as tall as it is wide breaks that, and the next measurement states a test sharp enough to fail at a single shell. If stretching the axis by 1/S turns the ellipse into a circle and T into T/S, then the ellipse’s boundary at translation T must be the circle’s at T/S exactly, and a bisection on the drawn outlines will find the first shell where it is not.